Minimum Moves to Make Array Complementary

Minimum Moves to Make Array Complementary

1. What is this problem about?

This "Medium/Hard" problem involves an array of length n (where n is even) and an integer limit. You want to make the array "complementary." An array is complementary if for all i from 0 to n/2 - 1, the sum nums[i] + nums[n - 1 - i] is equal to the same constant S.

You can replace any element in the array with any integer between 1 and limit. Each replacement counts as one move. Your goal is to find the minimum moves to make the array complementary for some S.

2. Why is this asked in interviews?

Companies like CureFit use this to test a candidate's ability to use Difference Arrays or Line Sweep algorithms. It requires:

• Range Analysis: Identifying for a pair (a, b), what range of S requires 0, 1, or 2 moves.
• Efficiency: Solving in $O(n + limit)$ time. A brute force checking every S would be $O(limit \times n)$, which is too slow.
• Mathematical Modeling: Turning a pair of numbers into a set of cost intervals.

3. Algorithmic pattern used

The pattern is Difference Array (Line Sweep). For each pair (a, b):

1. 2 moves: Required for any sum S in the range [2, 2 * limit].
2. 1 move: Required for any sum S in the range [min(a, b) + 1, max(a, b) + limit]. (This "overwrites" one of the 2-move costs).
3. 0 moves: Required if S == a + b. (This "overwrites" one of the 1-move costs).

By using a difference array to track the "change in moves" at specific boundaries, we can calculate the total moves for all possible S in one linear pass.

4. Example explanation

limit = 6, Pair (3, 4)

• Possible sums: 2 to 12.
• Default: 2 moves for everything.
• Range for 1 move: min(3,4)+1=4 to max(3,4)+6=10. In the range [4, 10], we only need 1 move.
• Specific sum for 0 moves: 3+4=7. At S=7, we need 0 moves. We repeat this for all pairs and find the S that has the lowest total moves.

5. Common mistakes candidates make

• Brute Force: Trying every possible sum S from 2 to 2 * limit and iterating through all pairs for each.
• Incorrect Range Boundaries: Forgetting that the smallest possible value is 1, so the smallest possible sum of two values is 2.
• Difference Array Logic: Not correctly updating the delta values at the boundaries of the ranges.

6. Interview preparation tip

The "Difference Array" is a powerful tool for problems involving "adding a value to a range." Whenever you have many ranges and you want to find a point that is covered by the most (or least) cost, think Line Sweep.